Royal biographer Andrew Morton, who wrote Diana: Her True Story, claimed: "Harry has sent presents to William and Kate and the calligraphy has been done by Meghan."
He also added that Meghan and Harry also received a present in return, as well as something for Archie.
Kensington Palace declined to comment.
Andrew Morton also went on to claim that Harry "plans to speak to his father and brother over Christmas".
This year's Christmas is set to look a lot different for the royal family, with Harry, Meghan and Archie in California, and the Cambridges celebrating alone as a family of five, due to government restrictions.
The Queen and Prince Philip have decided to spend the festive season "very quietly" at Windsor Castle in order to ensure all staff and members of the family remain safe during the pandemic.
The Queen and 99-year-old Prince Philip will remain protected at Windsor by "HMS Bubble" – a small number of staff who have isolated from their own families.
Special provision has been made to allow the Queen to attend church with a tiny service, to be held in her private chapel at the castle.